Bipolar electron waveguides in two-dimensional materials with tilted Dirac cones

Abstract

We show that the (2+1)-dimensional massless Dirac equation, which includes a tilt term, can be reduced to the biconfluent Heun equation for a broad range of scalar confining potentials, including the well-known Morse potential. Applying these solutions, we investigate a bipolar electron waveguide in 8-$Pmmn$ borophene, formed by a well and barrier, both described by the Morse potential. We demonstrate that the ability of two-dimensional materials with tilted Dirac cones to localize electrons in both a barrier and a well can be harnessed to create pseudogaps in their electronic spectrum. These pseudogaps can be tuned through varying the applied top-gate voltage. Potential opto-valleytronic and terahertz applications are discussed.

Figures (3)

• Figure 1: (a) Illustration of the proposed experimental setup and (b) the bipolar potential created by a well and barrier, as described by the Morse potential: $U\left(x\right)=-\mathrm{sgn}\left(x\right)U_{0}e^{-\left|x\right|/l}\left(1-e^{-\left|x\right|/l}\right)$.
• Figure 2: The energy spectrum of guided modes (for the $s=1$ valley) in a bipolar Morse potential, $U\left(x\right)=-\mathrm{sgn}\left(x\right)U_{0}e^{-\left|x\right|/l}\left(1-e^{-\left|x\right|/l}\right)$, of strength $V_0= l U_{0} / \hbar v_x= 10$, as a function of wavenumber along the waveguide in a tilted Dirac material, defined by parameters $v_x=0.86\,v_{\mathrm{F}}$, $v_y=0.69\,v_{\mathrm{F}}$ and $v_t=0.32\,v_{\mathrm{F}}$, where $v_{\mathrm{F}}$ is the Fermi velocity of graphene. Here, only the three lowest modes associated with the individual well (red lines) and the three highest modes associated with the individual barrier (blue lines) that form the bipolar waveguide are displayed. The grey dashed lines indicate the boundary at which the bound states merge with the continuum.
• Figure 3: The normalized wavefunctions of guided modes (for the $s=1$ valley) in a bipolar Morse potential, $U\left(x\right)=-\mathrm{sgn}\left(x\right)U_{0}e^{-\left|x\right|/l}\left(1-e^{-\left|x\right|/l}\right)$, of strength $V_0= l U_{0} / \hbar v_x= 10$, for $k_yl=2$ and energy: (a) $\widetilde{\varepsilon}=2.195$, (b) $\widetilde{\varepsilon}=1.658$, (c) $\widetilde{\varepsilon}=0.864$, (d) $\widetilde{\varepsilon}=0.624$, (e) $\widetilde{\varepsilon}=-0.169$, and (f) $\widetilde{\varepsilon}=-0.706$, in a tilted Dirac material defined by parameters $v_x=0.86\,v_{\mathrm{F}}$, $v_y=0.69\,v_{\mathrm{F}}$ and $v_t=0.32\,v_{\mathrm{F}}$. The solid red and blue lines correspond to real parts of $\psi_a$ and $\psi_b$, respectively, while the dashed lines correspond to their imaginary parts. The black line shows the electron density $\left|\Psi\right|^{2}=\left|\psi_{a}\right|^{2}+\left|\psi_{b}\right|^{2}$.